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While recent experiments on the spin Seebeck effect have revealed thedecisive role of the magnon contribution to the heat current $Q$ in hybridsystems containing thin ferromagnetic layers, the available acoustic mismatchtheory does not account for their magnetic properties. Here, we analyzetheoretically the heat transfer through an insulating ferromagnet (F)sandwiched between two insulators (I). Depending on the relation between the Fthickness, $d$, and the mean free path of phonons generated by magnons,$l_{ls}$, we reveal two qualitatively different regimes in the nonlinear heattransport through the F/I interfaces. Namely, in thick F layers the regime ofconventional "Joule" heating with $Q \propto T_s^4$ is realized, in which thedetailed structure of the F/I interfaces is inessential. Here $T_s$ is themagnon temperature. By contrast, in thin F layers with $d\ll l_{ls}$, most ofphonons emitted by magnons can leave F without being absorbed in its interior,giving rise to the \emph{magnon overheating} regime with $Q \propto T_s^m$ and$m\gtrsim7$. Conditions for the examination of both regimes and thedetermination of $T_s$ from experiments are discussed. The reported results arerelevant for the theoretical analysis of the spin Seebeck effect and thedevelopment of magnon-based spin caloritronic devices.

Role of magnons and the size effect in heat transport through an insulating ferromagnet/insulator interface